Decomposing Berge graphs and detecting balanced skew partitions

A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class or has some kind of decomposition. Then, Chudnovsky proved stronger theorems. One of them restricts the allowed decompositions to 2-joins and balanced skew partitions. We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an O(n)-time algorithm for the same problem restricted to Berge graphs. Our algorithm is not constructive: it certifies that a graph has a balanced skew partition if it has one. It relies on a new decomposition theorem for Berge graphs, that is more precise than the previously known theorems and implies them easily. Our theorem also implies that every Berge graph can be decomposed in a first step by using only balanced skew partitions, and in a second step by using only 2-joins. Our proof of this new theorem uses at an essential step one of the theorems of Chudnovsky. AMS Mathematics Subject Classification: 05C17, 05C75